3.360 \(\int x (a+b x)^n \left (c+d x^2\right )^3 \, dx\)
Optimal. Leaf size=282 \[ -\frac{5 a d^2 \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{3 d^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+6}}{b^8 (n+6)}-\frac{a \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left (a^2 d+b^2 c\right )^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{3 a d \left (a^2 d+b^2 c\right ) \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^8 (n+3)}+\frac{d \left (35 a^4 d^2+30 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^8 (n+4)}-\frac{7 a d^3 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^3 (a+b x)^{n+8}}{b^8 (n+8)} \]
[Out]
-((a*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^2*c + a^2*d)^2*(b
^2*c + 7*a^2*d)*(a + b*x)^(2 + n))/(b^8*(2 + n)) - (3*a*d*(b^2*c + a^2*d)*(3*b^2
*c + 7*a^2*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) + (d*(3*b^4*c^2 + 30*a^2*b^2*c*d
+ 35*a^4*d^2)*(a + b*x)^(4 + n))/(b^8*(4 + n)) - (5*a*d^2*(3*b^2*c + 7*a^2*d)*(a
+ b*x)^(5 + n))/(b^8*(5 + n)) + (3*d^2*(b^2*c + 7*a^2*d)*(a + b*x)^(6 + n))/(b^
8*(6 + n)) - (7*a*d^3*(a + b*x)^(7 + n))/(b^8*(7 + n)) + (d^3*(a + b*x)^(8 + n))
/(b^8*(8 + n))
_______________________________________________________________________________________
Rubi [A] time = 0.368504, antiderivative size = 282, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056
\[ -\frac{5 a d^2 \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+5}}{b^8 (n+5)}+\frac{3 d^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+6}}{b^8 (n+6)}-\frac{a \left (a^2 d+b^2 c\right )^3 (a+b x)^{n+1}}{b^8 (n+1)}+\frac{\left (a^2 d+b^2 c\right )^2 \left (7 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^8 (n+2)}-\frac{3 a d \left (a^2 d+b^2 c\right ) \left (7 a^2 d+3 b^2 c\right ) (a+b x)^{n+3}}{b^8 (n+3)}+\frac{d \left (35 a^4 d^2+30 a^2 b^2 c d+3 b^4 c^2\right ) (a+b x)^{n+4}}{b^8 (n+4)}-\frac{7 a d^3 (a+b x)^{n+7}}{b^8 (n+7)}+\frac{d^3 (a+b x)^{n+8}}{b^8 (n+8)} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
-((a*(b^2*c + a^2*d)^3*(a + b*x)^(1 + n))/(b^8*(1 + n))) + ((b^2*c + a^2*d)^2*(b
^2*c + 7*a^2*d)*(a + b*x)^(2 + n))/(b^8*(2 + n)) - (3*a*d*(b^2*c + a^2*d)*(3*b^2
*c + 7*a^2*d)*(a + b*x)^(3 + n))/(b^8*(3 + n)) + (d*(3*b^4*c^2 + 30*a^2*b^2*c*d
+ 35*a^4*d^2)*(a + b*x)^(4 + n))/(b^8*(4 + n)) - (5*a*d^2*(3*b^2*c + 7*a^2*d)*(a
+ b*x)^(5 + n))/(b^8*(5 + n)) + (3*d^2*(b^2*c + 7*a^2*d)*(a + b*x)^(6 + n))/(b^
8*(6 + n)) - (7*a*d^3*(a + b*x)^(7 + n))/(b^8*(7 + n)) + (d^3*(a + b*x)^(8 + n))
/(b^8*(8 + n))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 77.0045, size = 265, normalized size = 0.94 \[ - \frac{7 a d^{3} \left (a + b x\right )^{n + 7}}{b^{8} \left (n + 7\right )} - \frac{5 a d^{2} \left (a + b x\right )^{n + 5} \left (7 a^{2} d + 3 b^{2} c\right )}{b^{8} \left (n + 5\right )} - \frac{3 a d \left (a + b x\right )^{n + 3} \left (a^{2} d + b^{2} c\right ) \left (7 a^{2} d + 3 b^{2} c\right )}{b^{8} \left (n + 3\right )} - \frac{a \left (a + b x\right )^{n + 1} \left (a^{2} d + b^{2} c\right )^{3}}{b^{8} \left (n + 1\right )} + \frac{d^{3} \left (a + b x\right )^{n + 8}}{b^{8} \left (n + 8\right )} + \frac{3 d^{2} \left (a + b x\right )^{n + 6} \left (7 a^{2} d + b^{2} c\right )}{b^{8} \left (n + 6\right )} + \frac{d \left (a + b x\right )^{n + 4} \left (35 a^{4} d^{2} + 30 a^{2} b^{2} c d + 3 b^{4} c^{2}\right )}{b^{8} \left (n + 4\right )} + \frac{\left (a + b x\right )^{n + 2} \left (a^{2} d + b^{2} c\right )^{2} \left (7 a^{2} d + b^{2} c\right )}{b^{8} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x+a)**n*(d*x**2+c)**3,x)
[Out]
-7*a*d**3*(a + b*x)**(n + 7)/(b**8*(n + 7)) - 5*a*d**2*(a + b*x)**(n + 5)*(7*a**
2*d + 3*b**2*c)/(b**8*(n + 5)) - 3*a*d*(a + b*x)**(n + 3)*(a**2*d + b**2*c)*(7*a
**2*d + 3*b**2*c)/(b**8*(n + 3)) - a*(a + b*x)**(n + 1)*(a**2*d + b**2*c)**3/(b*
*8*(n + 1)) + d**3*(a + b*x)**(n + 8)/(b**8*(n + 8)) + 3*d**2*(a + b*x)**(n + 6)
*(7*a**2*d + b**2*c)/(b**8*(n + 6)) + d*(a + b*x)**(n + 4)*(35*a**4*d**2 + 30*a*
*2*b**2*c*d + 3*b**4*c**2)/(b**8*(n + 4)) + (a + b*x)**(n + 2)*(a**2*d + b**2*c)
**2*(7*a**2*d + b**2*c)/(b**8*(n + 2))
_______________________________________________________________________________________
Mathematica [B] time = 0.828552, size = 578, normalized size = 2.05 \[ \frac{(a+b x)^{n+1} \left (-5040 a^7 d^3+5040 a^6 b d^3 (n+1) x-360 a^5 b^2 d^2 \left (c \left (n^2+15 n+56\right )+7 d \left (n^2+3 n+2\right ) x^2\right )+120 a^4 b^3 d^2 (n+1) x \left (3 c \left (n^2+15 n+56\right )+7 d \left (n^2+5 n+6\right ) x^2\right )-6 a^3 b^4 d \left (3 c^2 \left (n^4+26 n^3+251 n^2+1066 n+1680\right )+30 c d \left (n^4+18 n^3+103 n^2+198 n+112\right ) x^2+35 d^2 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4\right )+6 a^2 b^5 d (n+1) x \left (3 c^2 \left (n^4+26 n^3+251 n^2+1066 n+1680\right )+10 c d \left (n^4+20 n^3+137 n^2+370 n+336\right ) x^2+7 d^2 \left (n^4+14 n^3+71 n^2+154 n+120\right ) x^4\right )-a b^6 \left (c^3 \left (n^6+33 n^5+445 n^4+3135 n^3+12154 n^2+24552 n+20160\right )+9 c^2 d \left (n^6+29 n^5+331 n^4+1871 n^3+5380 n^2+7172 n+3360\right ) x^2+15 c d^2 \left (n^6+25 n^5+241 n^4+1135 n^3+2734 n^2+3160 n+1344\right ) x^4+7 d^3 \left (n^6+21 n^5+175 n^4+735 n^3+1624 n^2+1764 n+720\right ) x^6\right )+b^7 \left (n^4+16 n^3+86 n^2+176 n+105\right ) x \left (c^3 \left (n^3+18 n^2+104 n+192\right )+3 c^2 d \left (n^3+16 n^2+76 n+96\right ) x^2+3 c d^2 \left (n^3+14 n^2+56 n+64\right ) x^4+d^3 \left (n^3+12 n^2+44 n+48\right ) x^6\right )\right )}{b^8 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8)} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x)^n*(c + d*x^2)^3,x]
[Out]
((a + b*x)^(1 + n)*(-5040*a^7*d^3 + 5040*a^6*b*d^3*(1 + n)*x - 360*a^5*b^2*d^2*(
c*(56 + 15*n + n^2) + 7*d*(2 + 3*n + n^2)*x^2) + 120*a^4*b^3*d^2*(1 + n)*x*(3*c*
(56 + 15*n + n^2) + 7*d*(6 + 5*n + n^2)*x^2) - 6*a^3*b^4*d*(3*c^2*(1680 + 1066*n
+ 251*n^2 + 26*n^3 + n^4) + 30*c*d*(112 + 198*n + 103*n^2 + 18*n^3 + n^4)*x^2 +
35*d^2*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4) + 6*a^2*b^5*d*(1 + n)*x*(3*c^2*
(1680 + 1066*n + 251*n^2 + 26*n^3 + n^4) + 10*c*d*(336 + 370*n + 137*n^2 + 20*n^
3 + n^4)*x^2 + 7*d^2*(120 + 154*n + 71*n^2 + 14*n^3 + n^4)*x^4) + b^7*(105 + 176
*n + 86*n^2 + 16*n^3 + n^4)*x*(c^3*(192 + 104*n + 18*n^2 + n^3) + 3*c^2*d*(96 +
76*n + 16*n^2 + n^3)*x^2 + 3*c*d^2*(64 + 56*n + 14*n^2 + n^3)*x^4 + d^3*(48 + 44
*n + 12*n^2 + n^3)*x^6) - a*b^6*(c^3*(20160 + 24552*n + 12154*n^2 + 3135*n^3 + 4
45*n^4 + 33*n^5 + n^6) + 9*c^2*d*(3360 + 7172*n + 5380*n^2 + 1871*n^3 + 331*n^4
+ 29*n^5 + n^6)*x^2 + 15*c*d^2*(1344 + 3160*n + 2734*n^2 + 1135*n^3 + 241*n^4 +
25*n^5 + n^6)*x^4 + 7*d^3*(720 + 1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5
+ n^6)*x^6)))/(b^8*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(8 +
n))
_______________________________________________________________________________________
Maple [B] time = 0.019, size = 1639, normalized size = 5.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x+a)^n*(d*x^2+c)^3,x)
[Out]
-(b*x+a)^(1+n)*(-b^7*d^3*n^7*x^7-28*b^7*d^3*n^6*x^7+7*a*b^6*d^3*n^6*x^6-3*b^7*c*
d^2*n^7*x^5-322*b^7*d^3*n^5*x^7+147*a*b^6*d^3*n^5*x^6-90*b^7*c*d^2*n^6*x^5-1960*
b^7*d^3*n^4*x^7-42*a^2*b^5*d^3*n^5*x^5+15*a*b^6*c*d^2*n^6*x^4+1225*a*b^6*d^3*n^4
*x^6-3*b^7*c^2*d*n^7*x^3-1098*b^7*c*d^2*n^5*x^5-6769*b^7*d^3*n^3*x^7-630*a^2*b^5
*d^3*n^4*x^5+375*a*b^6*c*d^2*n^5*x^4+5145*a*b^6*d^3*n^3*x^6-96*b^7*c^2*d*n^6*x^3
-7020*b^7*c*d^2*n^4*x^5-13132*b^7*d^3*n^2*x^7+210*a^3*b^4*d^3*n^4*x^4-60*a^2*b^5
*c*d^2*n^5*x^3-3570*a^2*b^5*d^3*n^3*x^5+9*a*b^6*c^2*d*n^6*x^2+3615*a*b^6*c*d^2*n
^4*x^4+11368*a*b^6*d^3*n^2*x^6-b^7*c^3*n^7*x-1254*b^7*c^2*d*n^5*x^3-25227*b^7*c*
d^2*n^3*x^5-13068*b^7*d^3*n*x^7+2100*a^3*b^4*d^3*n^3*x^4-1260*a^2*b^5*c*d^2*n^4*
x^3-9450*a^2*b^5*d^3*n^2*x^5+261*a*b^6*c^2*d*n^5*x^2+17025*a*b^6*c*d^2*n^3*x^4+1
2348*a*b^6*d^3*n*x^6-34*b^7*c^3*n^6*x-8592*b^7*c^2*d*n^4*x^3-50490*b^7*c*d^2*n^2
*x^5-5040*b^7*d^3*x^7-840*a^4*b^3*d^3*n^3*x^3+180*a^3*b^4*c*d^2*n^4*x^2+7350*a^3
*b^4*d^3*n^2*x^4-18*a^2*b^5*c^2*d*n^5*x-9420*a^2*b^5*c*d^2*n^3*x^3-11508*a^2*b^5
*d^3*n*x^5+a*b^6*c^3*n^6+2979*a*b^6*c^2*d*n^4*x^2+41010*a*b^6*c*d^2*n^2*x^4+5040
*a*b^6*d^3*x^6-478*b^7*c^3*n^5*x-32979*b^7*c^2*d*n^3*x^3-51432*b^7*c*d^2*n*x^5-5
040*a^4*b^3*d^3*n^2*x^3+3240*a^3*b^4*c*d^2*n^3*x^2+10500*a^3*b^4*d^3*n*x^4-486*a
^2*b^5*c^2*d*n^4*x-30420*a^2*b^5*c*d^2*n^2*x^3-5040*a^2*b^5*d^3*x^5+33*a*b^6*c^3
*n^5+16839*a*b^6*c^2*d*n^3*x^2+47400*a*b^6*c*d^2*n*x^4-3580*b^7*c^3*n^4*x-69936*
b^7*c^2*d*n^2*x^3-20160*b^7*c*d^2*x^5+2520*a^5*b^2*d^3*n^2*x^2-360*a^4*b^3*c*d^2
*n^3*x-9240*a^4*b^3*d^3*n*x^3+18*a^3*b^4*c^2*d*n^4+18540*a^3*b^4*c*d^2*n^2*x^2+5
040*a^3*b^4*d^3*x^4-4986*a^2*b^5*c^2*d*n^3*x-42360*a^2*b^5*c*d^2*n*x^3+445*a*b^6
*c^3*n^4+48420*a*b^6*c^2*d*n^2*x^2+20160*a*b^6*c*d^2*x^4-15289*b^7*c^3*n^3*x-746
28*b^7*c^2*d*n*x^3+7560*a^5*b^2*d^3*n*x^2-5760*a^4*b^3*c*d^2*n^2*x-5040*a^4*b^3*
d^3*x^3+468*a^3*b^4*c^2*d*n^3+35640*a^3*b^4*c*d^2*n*x^2-23706*a^2*b^5*c^2*d*n^2*
x-20160*a^2*b^5*c*d^2*x^3+3135*a*b^6*c^3*n^3+64548*a*b^6*c^2*d*n*x^2-36706*b^7*c
^3*n^2*x-30240*b^7*c^2*d*x^3-5040*a^6*b*d^3*n*x+360*a^5*b^2*c*d^2*n^2+5040*a^5*b
^2*d^3*x^2-25560*a^4*b^3*c*d^2*n*x+4518*a^3*b^4*c^2*d*n^2+20160*a^3*b^4*c*d^2*x^
2-49428*a^2*b^5*c^2*d*n*x+12154*a*b^6*c^3*n^2+30240*a*b^6*c^2*d*x^2-44712*b^7*c^
3*n*x-5040*a^6*b*d^3*x+5400*a^5*b^2*c*d^2*n-20160*a^4*b^3*c*d^2*x+19188*a^3*b^4*
c^2*d*n-30240*a^2*b^5*c^2*d*x+24552*a*b^6*c^3*n-20160*b^7*c^3*x+5040*a^7*d^3+201
60*a^5*b^2*c*d^2+30240*a^3*b^4*c^2*d+20160*a*b^6*c^3)/b^8/(n^8+36*n^7+546*n^6+45
36*n^5+22449*n^4+67284*n^3+118124*n^2+109584*n+40320)
_______________________________________________________________________________________
Maxima [A] time = 0.714953, size = 844, normalized size = 2.99 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n} c^{3}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac{3 \,{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} + \frac{3 \,{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} x^{6} +{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} x^{5} - 5 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} x^{4} + 20 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} x^{3} - 60 \,{\left (n^{2} + n\right )} a^{4} b^{2} x^{2} + 120 \, a^{5} b n x - 120 \, a^{6}\right )}{\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} + \frac{{\left ({\left (n^{7} + 28 \, n^{6} + 322 \, n^{5} + 1960 \, n^{4} + 6769 \, n^{3} + 13132 \, n^{2} + 13068 \, n + 5040\right )} b^{8} x^{8} +{\left (n^{7} + 21 \, n^{6} + 175 \, n^{5} + 735 \, n^{4} + 1624 \, n^{3} + 1764 \, n^{2} + 720 \, n\right )} a b^{7} x^{7} - 7 \,{\left (n^{6} + 15 \, n^{5} + 85 \, n^{4} + 225 \, n^{3} + 274 \, n^{2} + 120 \, n\right )} a^{2} b^{6} x^{6} + 42 \,{\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a^{3} b^{5} x^{5} - 210 \,{\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{4} b^{4} x^{4} + 840 \,{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{5} b^{3} x^{3} - 2520 \,{\left (n^{2} + n\right )} a^{6} b^{2} x^{2} + 5040 \, a^{7} b n x - 5040 \, a^{8}\right )}{\left (b x + a\right )}^{n} d^{3}}{{\left (n^{8} + 36 \, n^{7} + 546 \, n^{6} + 4536 \, n^{5} + 22449 \, n^{4} + 67284 \, n^{3} + 118124 \, n^{2} + 109584 \, n + 40320\right )} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n*x,x, algorithm="maxima")
[Out]
(b^2*(n + 1)*x^2 + a*b*n*x - a^2)*(b*x + a)^n*c^3/((n^2 + 3*n + 2)*b^2) + 3*((n^
3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*
b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n*c^2*d/((n^4 + 10*n^3 + 35*n^2 + 50*n
+ 24)*b^4) + 3*((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*b^6*x^6 + (n^5 +
10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a*b^5*x^5 - 5*(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^2
*b^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*a^3*b^3*x^3 - 60*(n^2 + n)*a^4*b^2*x^2 + 120*a
^5*b*n*x - 120*a^6)*(b*x + a)^n*c*d^2/((n^6 + 21*n^5 + 175*n^4 + 735*n^3 + 1624*
n^2 + 1764*n + 720)*b^6) + ((n^7 + 28*n^6 + 322*n^5 + 1960*n^4 + 6769*n^3 + 1313
2*n^2 + 13068*n + 5040)*b^8*x^8 + (n^7 + 21*n^6 + 175*n^5 + 735*n^4 + 1624*n^3 +
1764*n^2 + 720*n)*a*b^7*x^7 - 7*(n^6 + 15*n^5 + 85*n^4 + 225*n^3 + 274*n^2 + 12
0*n)*a^2*b^6*x^6 + 42*(n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*a^3*b^5*x^5 - 210*
(n^4 + 6*n^3 + 11*n^2 + 6*n)*a^4*b^4*x^4 + 840*(n^3 + 3*n^2 + 2*n)*a^5*b^3*x^3 -
2520*(n^2 + n)*a^6*b^2*x^2 + 5040*a^7*b*n*x - 5040*a^8)*(b*x + a)^n*d^3/((n^8 +
36*n^7 + 546*n^6 + 4536*n^5 + 22449*n^4 + 67284*n^3 + 118124*n^2 + 109584*n + 4
0320)*b^8)
_______________________________________________________________________________________
Fricas [A] time = 0.294262, size = 2261, normalized size = 8.02 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n*x,x, algorithm="fricas")
[Out]
-(a^2*b^6*c^3*n^6 + 33*a^2*b^6*c^3*n^5 + 20160*a^2*b^6*c^3 + 30240*a^4*b^4*c^2*d
+ 20160*a^6*b^2*c*d^2 + 5040*a^8*d^3 - (b^8*d^3*n^7 + 28*b^8*d^3*n^6 + 322*b^8*
d^3*n^5 + 1960*b^8*d^3*n^4 + 6769*b^8*d^3*n^3 + 13132*b^8*d^3*n^2 + 13068*b^8*d^
3*n + 5040*b^8*d^3)*x^8 - (a*b^7*d^3*n^7 + 21*a*b^7*d^3*n^6 + 175*a*b^7*d^3*n^5
+ 735*a*b^7*d^3*n^4 + 1624*a*b^7*d^3*n^3 + 1764*a*b^7*d^3*n^2 + 720*a*b^7*d^3*n)
*x^7 - (3*b^8*c*d^2*n^7 + 20160*b^8*c*d^2 + (90*b^8*c*d^2 - 7*a^2*b^6*d^3)*n^6 +
3*(366*b^8*c*d^2 - 35*a^2*b^6*d^3)*n^5 + 5*(1404*b^8*c*d^2 - 119*a^2*b^6*d^3)*n
^4 + 9*(2803*b^8*c*d^2 - 175*a^2*b^6*d^3)*n^3 + 2*(25245*b^8*c*d^2 - 959*a^2*b^6
*d^3)*n^2 + 24*(2143*b^8*c*d^2 - 35*a^2*b^6*d^3)*n)*x^6 - 3*(a*b^7*c*d^2*n^7 + 2
5*a*b^7*c*d^2*n^6 + (241*a*b^7*c*d^2 + 14*a^3*b^5*d^3)*n^5 + 5*(227*a*b^7*c*d^2
+ 28*a^3*b^5*d^3)*n^4 + 2*(1367*a*b^7*c*d^2 + 245*a^3*b^5*d^3)*n^3 + 20*(158*a*b
^7*c*d^2 + 35*a^3*b^5*d^3)*n^2 + 336*(4*a*b^7*c*d^2 + a^3*b^5*d^3)*n)*x^5 + (445
*a^2*b^6*c^3 + 18*a^4*b^4*c^2*d)*n^4 - 3*(b^8*c^2*d*n^7 + 10080*b^8*c^2*d + (32*
b^8*c^2*d - 5*a^2*b^6*c*d^2)*n^6 + (418*b^8*c^2*d - 105*a^2*b^6*c*d^2)*n^5 + (28
64*b^8*c^2*d - 785*a^2*b^6*c*d^2 - 70*a^4*b^4*d^3)*n^4 + (10993*b^8*c^2*d - 2535
*a^2*b^6*c*d^2 - 420*a^4*b^4*d^3)*n^3 + 2*(11656*b^8*c^2*d - 1765*a^2*b^6*c*d^2
- 385*a^4*b^4*d^3)*n^2 + 12*(2073*b^8*c^2*d - 140*a^2*b^6*c*d^2 - 35*a^4*b^4*d^3
)*n)*x^4 + 3*(1045*a^2*b^6*c^3 + 156*a^4*b^4*c^2*d)*n^3 - 3*(a*b^7*c^2*d*n^7 + 2
9*a*b^7*c^2*d*n^6 + (331*a*b^7*c^2*d + 20*a^3*b^5*c*d^2)*n^5 + (1871*a*b^7*c^2*d
+ 360*a^3*b^5*c*d^2)*n^4 + 20*(269*a*b^7*c^2*d + 103*a^3*b^5*c*d^2 + 14*a^5*b^3
*d^3)*n^3 + 4*(1793*a*b^7*c^2*d + 990*a^3*b^5*c*d^2 + 210*a^5*b^3*d^3)*n^2 + 560
*(6*a*b^7*c^2*d + 4*a^3*b^5*c*d^2 + a^5*b^3*d^3)*n)*x^3 + 2*(6077*a^2*b^6*c^3 +
2259*a^4*b^4*c^2*d + 180*a^6*b^2*c*d^2)*n^2 - (b^8*c^3*n^7 + 20160*b^8*c^3 + (34
*b^8*c^3 - 9*a^2*b^6*c^2*d)*n^6 + (478*b^8*c^3 - 243*a^2*b^6*c^2*d)*n^5 + (3580*
b^8*c^3 - 2493*a^2*b^6*c^2*d - 180*a^4*b^4*c*d^2)*n^4 + (15289*b^8*c^3 - 11853*a
^2*b^6*c^2*d - 2880*a^4*b^4*c*d^2)*n^3 + 2*(18353*b^8*c^3 - 12357*a^2*b^6*c^2*d
- 6390*a^4*b^4*c*d^2 - 1260*a^6*b^2*d^3)*n^2 + 72*(621*b^8*c^3 - 210*a^2*b^6*c^2
*d - 140*a^4*b^4*c*d^2 - 35*a^6*b^2*d^3)*n)*x^2 + 36*(682*a^2*b^6*c^3 + 533*a^4*
b^4*c^2*d + 150*a^6*b^2*c*d^2)*n - (a*b^7*c^3*n^7 + 33*a*b^7*c^3*n^6 + (445*a*b^
7*c^3 + 18*a^3*b^5*c^2*d)*n^5 + 3*(1045*a*b^7*c^3 + 156*a^3*b^5*c^2*d)*n^4 + 2*(
6077*a*b^7*c^3 + 2259*a^3*b^5*c^2*d + 180*a^5*b^3*c*d^2)*n^3 + 36*(682*a*b^7*c^3
+ 533*a^3*b^5*c^2*d + 150*a^5*b^3*c*d^2)*n^2 + 5040*(4*a*b^7*c^3 + 6*a^3*b^5*c^
2*d + 4*a^5*b^3*c*d^2 + a^7*b*d^3)*n)*x)*(b*x + a)^n/(b^8*n^8 + 36*b^8*n^7 + 546
*b^8*n^6 + 4536*b^8*n^5 + 22449*b^8*n^4 + 67284*b^8*n^3 + 118124*b^8*n^2 + 10958
4*b^8*n + 40320*b^8)
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x+a)**n*(d*x**2+c)**3,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.279571, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*(b*x + a)^n*x,x, algorithm="giac")
[Out]
Done